Graphics Reference
In-Depth Information
m
A
x
A
(
m
A
+
m
B
)
x
=
m
A
x
A
+
m
B
x
B
x
=
m
A
x
A
+
m
B
x
B
m
A
+
m
B
m
B
x
B
−
m
B
x
=
m
A
x
−
m
A
m
A
+
m
B
x
A
+
m
B
m
A
+
m
B
x
B
=
(11.7)
For example, if
m
A
=6and
m
B
= 12, and positioned at
x
A
=0and
x
B
=12
respectively, the centroid is located at
6
18
×
0+
12
x
=
18
×
12 = 8
Thus we can replace the two masses by a single mass of 18 located at
x
=8.
Note that the terms in (11.7)
m
A
/
(
m
A
+
m
B
)and
m
B
/
(
m
A
+
m
B
)sumto
1 and are identical to those used above for calculating ratios. They are also
called the
barycentric coordinates
of
x
relative to the points
A
and
B
.
Using the general form of (11.7) any number of masses can be analysed
using
i
=1
m
i
x
i
n
n
x
=
i
=1
m
i
where
m
i
is a mass located at
x
i
. Furthermore, we can compute the
y
-component of the centroid
y
using
i
=1
n
m
i
y
i
y
=
i
=1
n
m
i
and in 3D the
z
-component of the centroid
z
is
i
=1
m
i
z
i
n
n
z
=
i
=1
m
i
To recap, (11.7) states that
m
A
m
A
+
m
B
x
A
+
m
B
m
A
+
m
B
x
B
x
=
therefore, we can write
m
A
m
A
+
m
B
y
A
+
m
B
m
A
+
m
B
y
B
y
=
which allows us to state
m
A
m
A
+
m
B
A
+
m
B
m
A
+
m
B
B
P
=